How to perform PCA in the validation/test set?
for the first point I'm very sorry that I cannot give you any literature on this, but I might be able to explain you, why you don't take PCA on both datasets independently.

Principal components analysis is simply a transformation of your data into another (less dimensional) coordinate system. The axis for your new coordinate system are defined by the principal components (i.e. eigen-vectors) of your covariance matrix.

Since you will train your machine learning algorithm in the domain that is generated by the PCA, your test data must be exactly in the same domain. So as you said, you use exactly the same transformation for the test data as for the training data, i.e.
$y_{newdim} = y \phi $.
Of course if you applied standardization to your training data, you have to apply the same standardization to your test data. So you need to store the mean $\mu_x$ and the standard deviation $S_x$ and also standardize your test data y to 
\begin{equation}
y_{standardized} = \dfrac{y_i - \mu_x}{S_x}
\end{equation}

Note here that you have an error in your standardization formula (you do not need to divide by $M$).

The point is that the principal components of your test data $\phi_{y}$ would not match the principal components of your training data $\phi_{x}$. Thus the transformations from original space into PCA-space $\Phi_x(u)$ and $\Phi_y(u)$ would diverge and similar data points in the original space might be far away in the PCA-representation and vice versa. This is why you generate the mapping $x_{newdim} = \Phi_x(x) = x\phi$ and apply it also on the test data.

I hope I could make it clear to you.

Best